Hereditarily locally connected spaces

Question

Here, a space is called hereditarily locally connected ("hlc"), if each subset is locally connected.
Note that this is much stronger than the usual definition, which requires only connected subspaces to be locally connected. For instance, $\mathbb R$ fulfills the latter property, but is not hlc.

It is hard to find any papers dealing with hlc, since Google lists mainly examples for this much weaker property. Therefore, any references would be very welcome!

The only reference to hlc, I found, is this MO post, where it is also asked for references. Unfortunately, no answers.

My main question is:
Does there exist a non-discrete $T_2$ space, which is hlc?

Notes

1. Call a space partly connected, if all components are open. It easy to see, that a space is locally connected, iff all open subsets are partly connected. Hence, a space is hlc, iff all subsets are partly connected. This is helpful to prove the following:

2. There are many more examples of hlc spaces:
   Noetherian spaces, Alexandrov, filter like, hereditarily connected, arbitrary sums of hlc spaces, etc. But all these are discrete or not $T_2$.

3. Obviously, in an hlc space, every totally disconnected subset is discrete. But as far as I could see this is not very helpful, but I didn't find a T2 space with this weaker property, either.

4. In particular, an hlc $T_2$ space must not contain any non-trivial convergent sequence. Hence, if non-discrete, it cannot be sequential, first countable etc.

5. If it is even regular $T_2$ and hlc, any countable subset is closed discrete. But I couldn't even prove that regular $T_2$ hlc spaces do not exist. As pointed out by Jakobian, an hlc $T_2$ space does not contain any infinite countably compact subspace.

6. Since a locally connected, non-discrete space contains a non-discrete component, w.l.o.g we may assume that the space is connected.

7. A (perhaps) slightly weaker condition than hlc is "each closed subset is locally connected". I'm also not aware of any non-discrete $T_2$ example. There is definitely no compact $T_2$ example. See also this MO post.

Answer 1

My main question is:
Does there exist a non-discrete $T_2$ space, which is hlc?

I have a proof of the negative answer which looks too simple to be true. Indeed, suppose for a contradiction that $X$ is such a space and $x^*\in X$ be any nonisolated point. Let $x_0$ be any point of $X$ distinct from $x^*$. Since the space $X$ is Hausdorff, there exists an open neighborhood $U_0$ of $x_0$ such that $x^*\not\in\overline{U_0}$. Now we proceed by transfinite induction. At its $\alpha$-th step put $V_\alpha=X\setminus\overline{\bigcup_{\beta<\alpha} U_\beta}$. If $x^*\in V_\alpha$ then $V_\alpha$ is an open neighborhood of the point $x^*$. Since the point $x^*$ is nonisolated, there exists a point $x_\alpha\in V_\alpha$ distinct from $x^*$. Since the space $X$ is Hausdorff, there exists an open neighborhood $U_\alpha\subset V_\alpha$ of $x_\alpha$ such that $x^*\not\in\overline{U_\alpha}$.

When $x^*\not\in V_\alpha$ we are done. Put $Y=\{x^*\}\cup\bigcup_{\beta<\alpha} U_\beta$. The construction ensures that for each $\beta<\alpha$, the set $U_\beta$ is both open and closed subset of $Y$. For each neighborhood $U$ of $x^*$ in $Y$ there exists $\beta<\alpha$ such that the intersection $U\cap U_\beta$ is nonempty (and is proper, open and closed subset of $U$), so $U$ is not connected, a contradiction.